|
%I #25 Jul 05 2020 18:40:39
%S 3,21,48,91,133,243,273,336,481,768,651,741,931,1281,1875,1333,1456,
%T 1701,2128,2821,3888,2451,2613,2923,3441,4251,5461,7203,4161,4368,
%U 4753,5376,6321,7696,9633,12288,6643,6901,7371,8113,9211,10773,12931,15841,19683
%N Triangle read by rows: T(n,k) = n^4 + (n*k)^2 + k^4, 1 <= k <= n.
%C Entry 17a from July 9, 1796 in Gauss's Mathematical Diary: "Summa trium quadratorum continue proportionalium numquam primus esse potest: conspicuum exemplum novimus et quod congruum videtur. Confidamus." Paul Bachmann explains that this note is based on Gauss's discovery of this factorization: n^4 + n^2*k^2 + k^4 = (n^2 + n*k + k^2) * (n^2 - n*k + k^2).
%D Carl Friedrich Gauss (Hans Wussing, ed.), Mathematisches Tagebuch 1796-1814, Ostwalds Klassiker der Exakten Wissenschaften, Leipzig (1976, 1979), pp. 43, 63, 90.
%H Reinhard Zumkeller, <a href="/A219069/b219069.txt">Rows n = 1..120 of triangle, flattened</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Gauss's_diary">Gauss's diary</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Paul_Gustav_Heinrich_Bachmann">Paul Gustav Heinrich Bachmann</a>
%F T(n,k) = A215630(n,k) * A215631(n,k), 1 <= k <= n.
%e The triangle begins:
%e . 1: 3
%e . 2: 21 48
%e . 3: 91 133 243
%e . 4: 273 336 481 768
%e . 5: 651 741 931 1281 1875
%e . 6: 1333 1456 1701 2128 2821 3888
%e . 7: 2451 2613 2923 3441 4251 5461 7203
%e . 8: 4161 4368 4753 5376 6321 7696 9633 12288
%e . 9: 6643 6901 7371 8113 9211 10773 12931 15841 19683
%e . 10: 10101 10416 10981 11856 13125 14896 17301 20496 24661 30000
%e . 11: 14763 15141 15811 16833 18291 20293 22971 26481 31003 36741 43923
%t Table[n^4+(n*k)^2+k^4,{n,10},{k,n}]//Flatten (* _Harvey P. Dale_, Jul 05 2020 *)
%o (Haskell)
%o a219069 n k = a219069_tabl !! (n-1) !! (k-1)
%o a219069_row n = a219069_tabl !! n
%o a219069_tabl = zipWith (zipWith (*)) a215630_tabl a215631_tabl
%Y Cf. A059826 (left edge), A219056 (right edge), A219070 (row sums).
%Y Cf. A239426 (central terms).
%Y Cf. A243201 (diagonal (n + 1, n)). - _Mathew Englander_, Jun 03 2014
%K nonn,tabl,look
%O 1,1
%A _Reinhard Zumkeller_, Nov 11 2012
|
